By Brian Osserman
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Additional info for A Galois theory example
In  the solvability of sets of equations over Rmin,max and Rmax,min is treated. From the semiring theory point of view, it seems more natural to use the symbols + and x instead of EB and @ and, for consistency, 0 and 1 for the zero and the unit. However, in applications, often hybrid formulas are encountered containing conventional addition and multiplication as well as addition and multiplication in a semiring. For this reason, the notation for semirings will be carefully distinguished from that for operations in conventional algebra.
Such that v'f/ 1 i- c. Then [A® v]'f/ 1 = p, ® v'f/ 1 i- c. L ® v'f/2' implying that a'f/ 1 '112 i- c, v'f/2 i- c, and (TJ2,'r/1) E V(A). Following the same reasoning, a node 'r/3 can be found such that with a'f/2 '113 i- c, v'f/3 i- c, and (ry3, 'r/2) E V(A). Proceeding in this way, eventually some node, say, node 'r/h• must be encountered for a second time, because the number of nodes is finite. -2), ... L®l ® 0 v'f/h+k. k=O + in conventional algebra. -1 = l x p, + ~ v'f/h+k. 3), the average weight of the circuit '"Y then equals i"f/w /'"Y/1 We have thus proved the following lemma.
We came to talk about the well-known realization problem in mathematical systems theory. The simplest version is as follows. Rnx1, and C E IR 1 xn such that C A i B = g; for all i E N, everything in conventional algebra, and in such a way that n is as small as possible. The solution in conventional systems theory is well known, but what can one say about the same problem statement and set of equations in maxplus algebra? Professor Cuninghame-Green did not see the solution immediately. But when he left, thanking us for the hospitality, he seemed rather confident and said he would send us the solution the following week.